Write a real proof that absolute convergence implies convergence.
Assume that converges.
(a) Show that converges.
(b) Show that converges. (hint: an ≤ |an|)
(c) Show that converges.
(a) There's really nothing to do here. Since converges and multiplication by a constant doesn't affect convergence,
converges too. We could also say that since converges, so does
(b) Since ,
Since can't be less than zero (it could be equal, but not less than),
This is just what we need for the comparison test. We know from (a) that converges, so by the comparison test
(c) We know that and converge. This means we can take the difference of the series and get a new series that also converges:
This is what we wanted to show, so we're done.